SLAC-PUB-15129
QCD Analysis of the Scale-Invariance of Jets
Andrew J. Larkoski∗
SLAC National Accelerator Laboratory, Menlo Park, CA 94025
(Dated: July 9, 2012)
Abstract
Studying the substructure of jets has become a powerful tool for event discrimination and for
studying QCD. Typically, jet substructure studies rely on Monte Carlo simulation for vetting their
usefulness; however, when possible, it is also important to compute observables with analytic meth-
ods. Here, we present a global next-to-leading-log resummation of the angular correlation function
which measures the contribution to the mass of a jet from constituents that are within an angle R
with respect to one another. For a scale-invariant jet, the angular correlation function should scale
as a power of R. Deviations from this behavior can be traced to the breaking of scale invariance
in QCD. To do the resummation, we use soft-collinear effective theory relying on the recent proof
of factorization of jet observables at e+e− colliders. Non-trivial requirements of factorization of
the angular correlation function are discussed. The calculation is compared to Monte Carlo parton
shower and next-to-leading order results. The different calculations are important in distinct phase
space regions and exhibit that jets in QCD are, to very good approximation, scale invariant over
a wide dynamical range.
PACS numbers: 12.38.Bx,12.38.Cy,13.66.Bc,13.87.Fh
1
Submitted to Physical Review D
Work supported by US Department of Energy contract DE-AC02-76SF00515.
arXiv:1207.1437v1
I. INTRODUCTION
The current success of the Large Hadron Collider (LHC), its high center of mass energies,
its significant delivered integrated luminosity and its high-precision experiments has ushered
in a new era of particle physics. Particles and jets with significant transverse boosts are now
being copiously produced. An entire field of studying the substructure of highly boosted
jets has grown up out of the study of these objects and many methods have been proposed
to study QCD. In addition, procedures for discriminating QCD jets from jets initiated by
heavy particle decays have been introduced and new measurements of these methods are
being completed [1, 2]. To understand these methods in detail, most analyses have relied
on Monte Carlo simulation as the basis of study. However, Monte Carlo simulations have
limitations, and, where possible, it is vital to also compute the observables to higher orders
in QCD so as to have another handle on their behavior.
An important contribution to this effort of computing jet observables is resummation
of large logarithms that arise in fixed-order perturbation theory. Jets are objects that
are typically dominated by soft or collinear emissions and so it is necessary to resum the
logarithms that exist for an accurate prediction of an observable. Very recently, groups
have computed resummed contributions to light jet masses at hadron colliders [3] and N-
subjettiness [4, 5] in color-singlet jets at the LHC [6]. Ref. [6] in particular relied on the
factorization of color singlet processes at hadron colliders to reinterpret results from e+e−
colliders. Computing the resummed contribution to generic observables at hadron colliders is
made more difficult by the color flow throughout the collision which can destroy factorization.
To avoid discussion of these issues, here we will only consider jet observables at e+e− colliders.
In this paper, we will discuss the resummation of the angular correlation function introduced
in [7] using soft collinear effective theory (SCET) [8–11].
The angular correlation function G(R) was defined in [7] as
G(R) =∑
i 6=jp⊥ip⊥j∆R
2ijΘ(R−∆Rij) , (1)
for studying the substructure of jets at the LHC. ∆Rij is the boost-invariant angle between
particles i and j, the sum runs over all constituents of a jet and Θ is the Heaviside theta
function. The angular correlation function has distinct properties for scaleless jets versus jets
with at least one heavy mass scale. In particular, any structure in the angular correlation
2
function should be distributed roughly as RD, where D is a constant, for a scaleless jet.
It was shown that by exploiting the different behavior of the angular distribution of hard
structure in QCD jets versus jets initiated by heavy particle decay, an efficient tagging
algorithm could be defined.
Ref. [12] continued studying the properties of the angular correlation function, focusing
on average properties of QCD jets. It was shown through simple calculations that, for QCD
jets, the angular correlation function averaged over an ensemble of jets should approximately
scale as
〈G(R)〉 ' R2 , (2)
where the angle brackets are defined by
〈G(R)〉 =1
Njets
Njets∑
i=1
G(R)i . (3)
Deviations from R2 are due to the running coupling and higher order effects. The introduc-
tion of an ensemble averaged angular correlation function allows for a rigorous definition of
the dimension of a QCD jet which is also infrared and collinear (IRC) safe. This dimension
is defined to be the average angular structure function 〈∆G〉 and is the power to which the
average angular correlation function scales with R:
〈∆G(R)〉 ≡ d log〈G(R)〉d logR
. (4)
For QCD jets, 〈∆G〉 ∼ 2. In [12], it was also shown that the scaling of non-perturbative
physics in R is distinctly different, and this was used to determine the average energy density
of the underlying event.
Here, we will continue the work of [12] and compute the average angular structure function
by resummation within the context of SCET. Our analysis is only truly appropriate at e+e−
colliders, but we expect that the largest effect in going to hadron colliders is the contribution
of the underlying event. For this calculation, we introduce generalized correlation functions
Ga(R) parametrized by an index a:
Ga(R) =1
2E2J
∑
i 6=jEiEj sin θij tana−1
θij2
Θ(R− θij) (5)
The form of the angular correlation function is similar to jet angularity [13, 14]. However,
for our purposes here, we choose to index the parameter a such that the angular correlation
3
function is IRC safe for all a > 0. In the small angle limit, this reduces to Eq. 1 with a = 2
(up to normalization). The parameter a allows for a study of the behavior of the angular
correlation function with angular scales weighted differently. Analogously to the angular
structure function, we define a generalized average angular structure function
〈∆Ga〉 ≡d log〈Ga〉d logR
. (6)
The calculation and interpretation of average angular structure function will be the focus of
this paper.
In Sec. II, we discuss the factorization of jet observables in SCET and the computation
of the angular correlation function including global next-to-leading-log (NLL) contributions
for jets defined by a kT -type algorithm [15, 16]. The existence of a factorization theorem for
the angular correlation function is non-trivial. We will discuss the consistency conditions
that the angular correlation function satisfies for factorization. We will also briefly discuss
how the results obtained here can be used in a calculation of the angular correlation function
at the LHC. In Sec. III we compare the SCET calculation to a next-to-leading-order (NLO)
calculation of the angular correlation function. Resummation and fixed-order corrections
affect different parts of distributions and so the differences between the resummed calculation
and the fixed-order result give some sense as to the importance of these effects. This analysis
leads to Sec. IV, were we present a comparison between the SCET calculation and the output
of parton shower Monte Carlo. We observe significant differences between SCET and Monte
Carlo, but higher fixed order effects are substantial. We discuss some of the uncertainties in
the parton shower studying the effect of the evolution variable on the value of the angular
structure function. Finally, we present our conclusions in Sec. V.
II. SCET CALCULATION
SCET is an effective theory of QCD in which all modes of QCD are integrated out except
those corresponding to soft or collinear modes. Collinear and soft modes are defined by their
scaling with power counting parameter λ:
collinear ∼ (λ2, 1, λ) ,
soft ∼ (λ2, λ2, λ2) ,
4
which is the scaling of the +, − and transverse components of the momenta, respectively.
λ is a parameter that is defined for a particular process or observable; for example, for
computing the distribution of jet masses, λ ∼ mJ/p⊥J � 1. The fact that λ� 1 allows for
a systematic expansion in powers of λ. Higher order terms in λ are power suppressed (much
like the subleading terms in the twist expansion).
For an event shape observable O that factorizes, the cross section can be written in the
schematic form:
dσ
dO = H(µ)
[∏
ni
Jni(O;µ)
]⊗ S(O;µ) , (7)
where H(µ) is the hard function, which matches the full QCD result at a scale µ, J(µ;ni,O)
is the jet function for the contribution to the observable O from ni-collinear modes and
S(µ;O) is the soft function for the contribution to the observable O from the soft modes.
⊗ represents a convolution between the jet and soft functions. All functions depend on the
factorization scale µ.
Factorization of jet observables in SCET was first exhibited in [17, 18]. Ref. [18] computed
individual jet angularities to NLL in e+e− collisions. It was shown that factorization of the
cross section for jet observables in e+e− → N jets has the form
dσ
dO1 · · · dOM= H(n1, . . . , nN ;µ)
[M∏
i=1
Jni(Oi;µ)
]⊗ Sn1···nN
(O1, . . . ,OM ;µ)N∏
j=M+1
J(µ) ,(8)
where M ≤ N of the jet observables Oi have been measured. Jet directions are denoted by
ni and J(Oi;µ) is the jet function for a jet in which the observable Oi has been measured
and J(µ) is a jet function for a jet which has not been measured. We will refer to these as
the measured and unmeasured jet functions, respectively. A similar nomenclature will be
used for the soft functions. Jet algorithm dependence and jet energies have been suppressed.
An important point from [18] is that factorization requires that the jets be well-separated;
namely, that
tij =tan
ψij
2
tan R0
2
� 1 , (9)
where ψij the is angle between any pair of jets i, j and R0 is the jet algorithm radius. We will
assume that this condition is met in the following and leave any discussion of subtleties to
[18]. A non-trivial requirement of the factorization is the independence of the cross section
on the factorization scale µ. This requirement leads to a constraint that the sum of the
5
anomalous dimensions of the hard, jet and soft functions is zero. We will show that this
holds for the angular correlation function.
We will use the results of the factorization theorem proven in [17, 18] to compute the
distribution of the angular correlation function from Eq. 5. In particular, we are interested
in the ensemble average of the angular structure function as defined in Eq. 6. Note that this
observable is independent of any normalization factor of the angular correlation function;
thus, with the goal of computing the average angular structure function, it is consistent to
ignore factors that are independent of Ga and the angular resolution parameter R. Thus,
for the purposes of this paper, we can ignore the overall factors in the factorized form of
the cross section of the hard function and the unmeasured jet functions. In this case, the
factorized form of the cross section becomes
dσ
dGa1 · · · dGaM= C(µ)
[M∏
i=1
Jni(Gai;µ)
]⊗ Sn1···nN
(Ga1, . . . ,GaM ;µ) , (10)
where C(µ) is independent of Ga and the resolution parameter R.
In this section, we present a calculation of the jet and soft functions for the angular
correlation function for jets defined by a kT algorithm. We first argue that the angular
correlation function is computable in SCET and relate its form at NLO to the form of jet
angularity at NLO. This comparison will allow us to relate the calculation of the angular
correlation function to the work in [18]. We then present a calculation of the measured jet
and soft functions of the angular correlation function. From these results, we can determine
the anomalous dimensions of the jet and soft functions and will show the consistency of the
factorization relies on a non-trivial cancellation of dependence on the angular resolution R
between the jet and soft functions. We can then resum up to the next-to-leading logs of
the jet and soft functions by the renormalization group. Note that we do not attempt to
resum non-global logs [19] that arise due to the non-trivial phase space constraints of the
jet algorithm or the angular correlation function. From the resummed expression of the
angular correlation function, we find the ensemble average and compute the average angular
structure function numerically.
It should be stressed that non-global logarithms are ignored in this study. The angular
correlation function for a jet requires several phase space constraints; the jet algorithm, soft
jet vetoes, the resolution parameter R, etc. These provide numerous sources for non-global
logarithms which cannot be resummed analytically. The study of non-global logarithms in
6
QCD cross sections is a subtle and evolving story. For recent work in this direction, especially
in the context of non-global logarithms from jet clustering see, for example, [20–25]. It is
outside the scope of this paper to discuss non-global logarithms further.
A. Factorization of the Angular Correlation Function
Factorization of jet observables requires that soft modes only resolve the entire jet and
not individual collinear modes contributing to the jet. Angularity τa is a one-parameter
family of observables defined as [13, 14]
τa =1
2EJ
∑
i∈Je−ηi(1−a)p⊥i , (11)
where J is the jet, p⊥i is the momentum of particle i transverse to the jet axis and ηi is the
rapidity of particle i with respect to the jet axis:
ηi = − log tanθi2. (12)
Angularity is IRC safe for a < 2. The separation of soft and collinear modes in angularity
is simple to show. To leading power in λ,
τa =1
2EJ
∑
C∈Je−ηC(1−a)p⊥C +
1
2EJ
∑
S∈Je−ηS(1−a)p⊥S
= τCa + τSa , (13)
where C and S represent the collinear and soft modes, respectively. Note that the soft
modes do not affect the location of the jet center to leading power in λ. Factorization of
angularities exists only for a < 1 due to the presence of logarithms of rapidity; however,
recently it was shown that these logarithms can be controlled [26, 27]. We will show that
angularity and the angular correlation function have similarities which will allow us to use
many of the results from [18] here.
To justify the use of SCET for computing the angular correlation function, we must first
show that the angular correlation function does not mix soft and collinear modes. This
argument was presented in [12] (based on arguments from [28]), but we present it here for
completeness. In terms of soft and collinear modes, the angular correlation function can be
7
expressed as
Ga(R) =1
2E2J
∑
i 6=jEiEj sin θij tana−1
θij2
Θ(R− θij)
=1
2E2J
∑
i,j∈CEiEj sin θij tana−1
θij2
Θ(R− θij)
+1
2E2J
∑
i,j∈SEiEj sin θij tana−1
θij2
Θ(R− θij)
+1
2E2J
∑
C,S
ECES sin θCS tana−1θCS2
Θ(R− θCS) . (14)
Note that, to NLO, there is no soft-soft correlation contribution to the angular correlation
function because such a term would require the radiation of two soft gluons which first occurs
at NNLO. To accuracy of the leading power in λ, we can exchange the collinear modes with
the jet itself in the collinear-soft term. Explicitly,
θCS = θJS +O(λ) , (15)
as the angle of the soft modes with respect to the jet center scales as θJS ∼ 1. Appropriate
for NLO or NLL, the angular correlation function can be written as
Ga(R) =1
2E2J
∑
i,j∈CEiEj sin θij tana−1
θij2
Θ(R− θij)
+1
2EJ
∑
S
ES sin θJS tana−1θJS2
Θ(R− θJS) . (16)
Thus, the collinear and soft modes are decoupled to leading power and so the angular
correlation function is factorizable, and hence computable, in SCET.
To NLO, a jet is composed of at most two particles, so the form of many observables
simplifies substantially at this order. The form of the angular correlation function from Eq. 5
was chosen so as to be similar in form to angularity. The contribution to the angularity and
the angular correlation function from collinear modes is distinct. The measured jet functions
will need to be recomputed for the angular correlation function. However, the contributions
to the angularity and the angular correlation function from soft modes are simply related:
GSa (R) =ES2EJ
sin θJS tana−1θJS2
Θ(R− θJS) = τS2−aΘ(R− θJS) . (17)
This observation will allow us to recycle the soft function calculation for angularity for the
angular correlation function.
8
An important point to note here is that the scaling of the angle between collinear modes
i and j goes like θij ∼ λ. Thus, to leading power, the angular correlation function for the
collinear-collinear contribution can be written as
GCCa =1
2E2J
∑
i,j∈CEiEj sin θij tana−1
θij2
Θ(R− θij)
=1
E2J
∑
i,j∈CEiEj tana
θij2
Θ(R− θij) . (18)
We will use this form of the collinear-collinear contribution to the angular correlation func-
tion for computing the measured jet functions.
B. Measured Jet Functions
The leading power contribution to the measured jet functions at NLO comes from two
collinear particles which are clustered in the jet and can be computed from cutting one-
loop SCET diagrams. The phase space integrals can be extended over the entire range of
momentum for the collinear particles in the jet as long as the contribution from the zero
momentum bin is subtracted [29]. In particular, we consider a jet with light cone momentum
l = (l+, ω, 0) which splits to two collinear particles with light cone momenta q = (q+, q−,q⊥)
and l− q = (l+ − q+, ω − q−,−q⊥). The zero-bin subtraction term can be determined from
the measured jet function by taking the scaling q ∼ λ2. We will refer to contribution to the
jet function that does not include the zero-bin subtraction as the naıve contribution.
To compute the measured jet function, we will need to enforce phase space cuts from
the jet algorithm and the observable. We will compute the jet function for a kT -type jet
algorithm as defined by a jet radius R0. At NLO, all kT algorithms are the same and two
particles are clustered in the jet if their angular separation is less than R0. This leads to
the phase space constraint
ΘkT = Θ
(cosR0 −
q · (l− q)
|q|√
(l− q)2
)= Θ
(tan2 R0
2− q+ω2
q−(ω − q−)2
), (19)
where on the right, the leading scaling behavior was kept. The jet algorithm constraint for
the zero-bin subtraction term is then
Θ(0)kT
= Θ
(tan2 R0
2− q+
q−
). (20)
9
The phase space constraints for the angular correlation function are more subtle. The
δ-function which constrains a jet to have angular correlation function Ga, δR = δ(Ga − Ga),is
δR = δ
(Ga − ωa−2(ω − q−)1−a(q−)1−a/2(q+)a/2Θ
(tan2 R
2− q+ω2
q−(ω − q−)2
)), (21)
where R is the resolution parameter of the angular correlation function. For a kT -type jet at
NLO, the angular correlation function vanishes if R > R0; thus, we will assume that R < R0
in the following. This δ-function can be decomposed depending on the value of Θ-function
as
δR = δ
(Ga − ωa−2(ω − q−)1−a(q−)1−a/2(q+)a/2Θ
(tan2 R
2− q+ω2
q−(ω − q−)2
))
= δ(Ga − ωa−2(ω − q−)1−a(q−)1−a/2(q+)a/2
)Θ
(tan2 R
2− q+ω2
q−(ω − q−)2
)
+ δ (Ga) Θ
(q+ω2
q−(ω − q−)2− tan2 R
2
). (22)
The δ-function for the zero-bin subtraction term is found by taking q ∼ λ2:
δ(0)R = δ
(Ga − ω−1(q−)1−a/2(q+)a/2
)Θ
(tan2 R
2− q+
q−
)+ δ (Ga) Θ
(q+
q−− tan2 R
2
). (23)
1. Measured Quark Jet Function
The naıve contribution to the measured quark jet function can be computed in dimen-
sional regularization from the diagrams shown in Fig. 1:
Jqω(Ga) = g2µ2εCF
∫dl+
2π
1
(l+)2
∫ddq
(2π)d
(4l+
q−+ (d− 2)
l+ − q+ω − q−
)
×2πδ(q+q− − q2⊥
)Θ(q−)Θ(q+)2πδ
(l+ − q+ − q2⊥
ω − q−)
×Θ(ω − q−)Θ(l+ − q+)Θ
(tan2 R0
2− q+ω2
q−(ω − q−)2
)
×[δ(Ga − ωa−2(ω − q−)1−a(q−)1−a/2(q+)a/2
)Θ
(tan2 R
2− q+ω2
q−(ω − q−)2
)
+ δ (Ga) Θ
(q+ω2
q−(ω − q−)2− tan2 R
2
)]. (24)
We take d = 4− 2ε. The coefficient to the δ(Ga) term can be found by integrating over Ga.The terms that remain are +-distributions, which integrate to zero. The zero-bin subtraction
10
(B)(A) (D)(C)(A) (A)
Figure 4: Diagrams contributing to the quark jet function. (A) and (B) Wilson line emission
diagrams; (C) and (D) QCD-like diagrams. The momentum assignments are the same as in Fig. 3.
The zero bin of particle 2 is given by the replacement q ! l � q.
For all the jet algorithms we consider, the zero-bin subtractions of the unmeasured jet
functions are scaleless integrals.12 However, for the measured jet functions, the zero-bin
subtractions give nonzero contributions that are needed for the consistency of the e↵ective
theory.
In the case of a measured jet, in addition to the phase space restrictions we also demand
that the jet contributes to the angularity by an amount ⌧a with the use of the delta function
�R = �(⌧a � ⌧a), which is given in terms of q and l by
�R ⌘ �R(q, l+) = �
✓⌧a �
1
!(! � q�)a/2(l+ � q+)1�a/2 � 1
!(q�)a/2(q+)1�a/2
◆. (4.4)
In the zero-bin subtraction of particle 1, the on-shell conditions can be used to write the
corresponding zero-bin �-function as
�(0)R = �
✓⌧a �
1
!(q�)a/2(q+)1�a/2
◆, (4.5)
(and for particle 2 with q ! l � q).
4.2 Quark Jet Function
The diagrams corresponding to the quark jet function are shown in Fig. 4. The fully
inclusive quark jet function is defined as
Zd4x eil·x h0|�a↵
n,!(x)�b�n,!(0) |0i ⌘ �ab
✓n/
2
◆↵�Jq!(l+) , (4.6)
and has been computed to NLO (see, e.g., [75, 76]) and to NNLO [77]. Below we compute
the quark jet function at NLO with phase space cuts for the jet algorithm for both the
measured jet, Jq!(⌧a), and the unmeasured jet, Jq
!. As discussed above, we will find that
the only nonzero contributions come from cuts through the loop when both cut particles
are inside the jet.
12Note that algorithms do exist that give nonzero zero-bin contributions to unmeasured jet functions [32].
– 32 –
FIG. 1. SCET Feynman diagrams contributing to the quark jet function.
term follows from taking the scaling limit q ∼ λ2 of the naıve jet function above:
Jq(0)ω (Ga) = 4g2µ2εCF
∫dl+
2π
1
l+
∫ddq
(2π)d1
q−2πδ
(q+q− − q2⊥
)Θ(q−)Θ(q+)
×2πδ(l+ − q+
)Θ(l+ − q+)Θ
(tan2 R0
2− q+
q−
)
×[δ(Ga − ω−1(q−)1−a/2(q+)a/2
)Θ
(tan2 R
2− q+
q−
)
+ δ (Ga) Θ
(q+
q−− tan2 R
2
)]. (25)
The term proportional to δ(Ga) is scaleless and integrates to zero in pure dimensional regu-
lation.
Employing a MS scheme, we find the measured quark jet function for kT -type jet algo-
rithms of Ga to be
Jqω(Ga) = Jqω(Ga)− Jq(0)ω (Ga) =αsCF
2π
[(a
a− 1
1
ε2+
3
2
1
ε+
a
a− 1
log µ2
ω2
ε+
1
εlog
tan2 R2
tan2 R0
2
)δ(Ga)
− 2
a− 1
1
ε
(Θ(Ga)Ga
)
+
]+ Jqω(Ga, ε0) , (26)
where Jqω(Ga, ε0) consists of terms that are finite as ε → 0. These terms are presented in
Appendix A. The definition of the +-distribution is also given in Appendix A. Note that
the 1/ε terms for the angular correlation function are the same as those for angularity from
[18] with a → 2 − a plus an additional term of the logarithm of the ratio of scales; the
resolution scale R and the jet radius R0. This term contributes to the anomalous dimension
of the jet function. In principle, these logarithms could be attempted to be resummed.
However, note that the resolution scale R can never practically be parametrically smaller
than the jet radius R0, so these logarithms never become large. Thus, we will not worry
about resumming these logarithms.
11
Figure 5: Diagrams contributing to the gluon jet function. (A) sunset and (B) tadpole gluon
loops; (C) ghost loop; (D) sunset and (E) tadpole collinear quark loops; (F) and (G) Wilson line
emission loops. Diagrams (F) and (G) each have mirror diagrams (not shown). The momentum
assignments are the same as in Fig. 3.
Without inserting any additional constraints, this integral is scaleless and zero in dimen-
sional regularization. Therefore, in the absence of phase-space restrictions, the naıve inte-
gral Eq. (4.19) gives the standard (inclusive) gluon jet function
Jg!(l+)
2⇡!=↵s
4⇡µ2✏(!l+)�1�✏
TRNf
✓4
3+
20
9✏
◆� CA
✓4
✏+
11
3+
✓67
9� ⇡2
◆✏
◆�, (4.21)
in the MS scheme. The measured and unmeasured jet functions are obtained by inserting
⇥alg�R and ⇥alg, respectively, into Eqs. (4.19) and (4.20).
4.3.1 Measured Gluon Jet
The naive contribution to the measured gluon jet can be written as
Jg!(⌧a) =
↵s
2⇡
1
�(1 � ✏)
✓4⇡µ2
!2
◆✏1
1 � a2
✓1
⌧a
◆1+ 2✏2�aZ 1
0dx (xa�1 + (1 � x)a�1)
2✏2�a (4.22)
⇥TRNf
✓1 � 2
1 � ✏x(1 � x)
◆� CA
✓2 � 1
x(1 � x)� x(1 � x)
◆�⇥alg(x) ,
where x ⌘ q�/!. This gives
Jg!(⌧a) =
↵s
2⇡
1
�(1 � ✏)
4⇡µ2
!2 tan2 R2
!✏�(⌧a)
"CA
✓1
✏2+
11
6
1
✏
◆� 2
3✏TRNf
#+↵s
2⇡Jg
alg(⌧a) ,
(4.23)
where, as for the quark jet function, the finite distributions Jgalg(⌧a) di↵er among the
algorithms we consider. They are given in Appendix A.
The zero-bin result is
Jg(0)! (⌧a) =
↵sCA
⇡
1
�(1 � ✏)
4⇡µ2 tan2(1�a) R
2
!2
!✏✓1
⌧a
◆1+2✏ 1
(1 � a)✏. (4.24)
– 36 –
FIG. 2. SCET Feynman diagrams contributing to the gluon jet function. Diagrams (F) and (G)
have mirrored counterparts which are not shown.
2. Measured Gluon Jet Function
The naıve contribution to the measured gluon jet function can be computed from the
diagrams shown in Fig. 2:
Jgω(Ga) = 2g2µ2ε
∫dl+
2π
1
l+
∫ddq
(2π)d1
ω − q−2πδ(q+q− − q2⊥
)2πδ
(l+ − q+ − q2⊥
ω − q−)
×{nFTR
(1− 2
1− εq+q−
ωl+
)− CA
(2− ω
q−− ω
ω − q− −q+q−
ωl+
)}
×Θ(q−)Θ(q+)Θ(ω − q−)Θ(l+ − q+)Θ
(tan2 R0
2− q+ω2
q−(ω − q−)2
)
×[δ(Ga − ωa−2(ω − q−)1−a(q−)1−a/2(q+)a/2
)Θ
(tan2 R
2− q+ω2
q−(ω − q−)2
)
+ δ (Ga) Θ
(q+ω2
q−(ω − q−)2− tan2 R
2
)]. (27)
The coefficient of the δ(Ga) term can be found by integrating over Ga. The terms that
remain are +-distributions, which integrate to zero. The zero-bin subtraction term follows
from taking the scaling limit l − q ∼ q ∼ λ2 of the naıve jet function above:
Jg(0)ω (Ga) = 4g2µ2εCA
∫dl+
2π
1
l+
∫ddq
(2π)d1
q−2πδ
(q+q− − q2⊥
)Θ(q−)Θ(q+)
×2πδ(l+ − q+
)Θ(l+ − q+)Θ
(tan2 R0
2− q+
q−
)
×[δ(Ga − ω−1(q−)1−a/2(q+)a/2
)Θ
(tan2 R
2− q+
q−
)
+ δ (Ga) Θ
(q+
q−− tan2 R
2
)]. (28)
The term proportional to δ(Ga) integrates to zero in pure dimensional regulation. This
zero-bin subtraction term is exactly the same up to color factors as the quark jet function
12
zero-bin subtraction.
Employing a MS scheme, we find the measured gluon jet function for the kT -type jet
algorithms of Ga to be
Jgω(Ga) = Jgω(Ga)− Jg(0)ω (Ga) =αs2π
[(CA
a
a− 1
1
ε2+β02ε
+ CAa
a− 1
log µ2
ω2
ε
+CAε
logtan2 R
2
tan2 R0
2
)δ(Ga)−
2CAε(a− 1)
(Θ(Ga)Ga
)
+
]
+ Jgω(Ga, ε0) , (29)
where Jgω(Ga, ε0) consists of terms that are finite as ε → 0. These terms are presented in
Appendix A. β0 is the coefficient of the one-loop β-function:
β0 =11
3CA −
2
3NF , (30)
with TR = 12. As with the quark jet function, the 1/ε terms are the same as those for
angularity from [18] with a→ 2− a plus an additional term of the logarithm of the ratio of
the resolution parameter R to the jet radius R0.
C. Measured Soft Function
As shown above, there is a simple relationship between the form of angularity for soft
modes and the angular correlation for soft modes. This relationship will allow us to use
the results from [18] in computing the measured soft function for the angular correlation
function. First, we consider the phase space constraints from the jet algorithm and the
angular correlation function. For the kT jet algorithm, soft radiation must be within the jet
radius R0 of the jet axis to be included:
ΘkT = Θ
(tan2 R0
2− k+
k−
). (31)
The δ-function that constrains the soft modes to contribute an amount Ga to the angular
correlation function is
δR = δ
(Ga − ω−1(k−)1−a/2(k+)a/2Θ
(tan2 R
2− k+
k−
))
= δ(Ga − ω−1(k−)1−a/2(k+)a/2
)Θ
(tan2 R
2− k+
k−
)+ δ (Ga) Θ
(k+
k−− tan2 R
2
). (32)
13
The measured soft function of a gluon emitted from lines i and j into a jet is
Smeasij (Ga) = −g2µ2εTi ·Tj
∫ddk
(2π)dni · nj
(ni · k)(nj · k)2πδ(k2)Θ(k0)ΘkT δR
= −g2µ2εTi ·Tj
∫ddk
(2π)dni · nj
(ni · k)(nj · k)2πδ(k2)Θ(k0)Θ
(tan2 R0
2− k+
k−
)
×[δ(Ga − ω−1(k−)1−a/2(k+)a/2
)Θ
(tan2 R
2− k+
k−
)
+ δ (Ga) Θ
(k+
k−− tan2 R
2
)]. (33)
Note that the integral proportional to δ(Ga) is scaleless and so vanishes in pure dimensional
regularization. Also, the integral is only non-zero if R < R0 and so the Θ-function from the
jet algorithm is redundant. Thus, we can write the soft function as
Smeasij (Ga) = −g2µ2εTi ·Tj
∫ddk
(2π)dni · nj
(ni · k)(nj · k)2πδ(k2)Θ(k0)Θ
(tan2 R
2− k+
k−
)
× δ(Ga − ω−1(k−)1−a/2(k+)a/2
). (34)
This is the same form of the measured soft function as for angularity with a jet radius equal
to R which was computed in [18]. Up to terms that are suppressed by 1/t2 from Eq. 9, the
measured soft function for jet i is
Smeas(Gia) = −αs2π
T2i
1
a− 1
{[1
ε2+
1
εlog
µ2 tan2(a−1) R2
ω2− π2
12+
1
2log2 µ
2 tan2(a−1) R2
ω2
]δ(Gia)
−2
[(1
ε+ log
µ2 tan2(a−1) R2
G2iaω2
)Θ(Gia)Gia
]
+
}, (35)
where T2i is the square of the color in the jet.
D. Anomalous Dimensions and Consistency Conditions
A non-trivial requirement of the factorization is that the physical cross section should
be independent of the factorization scale µ. A consequence of this is that the anomalous
dimensions of the hard, jet and soft functions must sum to 0. The requirement is
0 =
(γH(µ) + γunmeas
S (µ) +∑
i/∈meas
γJi(µ)
)δ(Ga) +
∑
i∈meas
(γJi(Gia;µ) + γmeas
S (Gia;µ)), (36)
where γH , γS and γJ are the anomalous dimensions of the hard, soft and jet functions. The
µ dependence must be summed over the measured and unmeasured jet and soft functions.
14
The sum of the hard, unmeasured soft and unmeasured jet anomalous dimensions to NLO
is
γH(µ) + γunmeasS (µ) +
∑
i/∈meas
γJi(µ) = −αsπ
∑
i∈meas
T2i log
µ2
ω2i tan2 R0
2
−∑
i∈meas
γi , (37)
where γi depends on the flavor of the jet:
γq =3αs2π
CF , γg =αsπ
11CA − 2NF
6=αs2πβ0 , (38)
for quark and gluon jets respectively. We will show that the measured jet and soft function
anomalous dimensions for the angular correlation function are exactly what is required to
satisfy Eq. 36.
The anomalous dimensions of the measured jet or soft functions are given by the coefficient
of the 1/ε terms from Eqs. 26, 29 and 35. The anomalous dimensions of the quark and gluon
jet functions can be written collectively as
γJi(Gia) =
[αsπT2i
(a
a− 1log
µ2
ω2i
+ logtan2 R
2
tan2 R0
2
)+ γi
]δ(Ga)− 2
αsπT2i
1
a− 1
[Θ(Ga)Ga
]
+
,
(39)
where γi is defined in Eq. 38. Note the non-trivial dependence of the anomalous dimension
on both the jet radius and the resolution parameter of the angular correlation function. The
anomalous dimension of the measured soft function for a quark or gluon jet is
γmeasS (Gia) = −αs
πT2i
1
a− 1
[δ(Ga) log
µ2 tan2(a−1) R2
ω2i
− 2
(Θ(Ga)Ga
)
+
]. (40)
As mentioned earlier, jet angularity is not factorizable for a = 1 and here we see that the
anomalous dimensions of the angular correlation jet and soft functions become meaningless
for a = 1, signaling a breakdown of factorization. For the angular correlation function, we
are most interested in a = 2, so we will not consider this issue further here.
Summing over the measured jet and soft function anomalous dimensions, we find
∑
i∈meas
(γJi(Gia;µ) + γmeas
S (Gia;µ))
=
(αsπ
∑
i∈meas
T2i log
µ2
ω2i tan2 R0
2
+∑
i∈meas
γi
)δ(Ga) (41)
Note that there is a non-trivial cancellation of the angular correlation function resolution
parameter R between the jet and soft functions. This contribution exactly cancels that from
the hard and unmeasured jet and soft functions in Eq. 37, consistent with the factorization
requirement.
15
E. Resummation and Averaging
To proceed with the resummation to NLL of the jet and soft functions, we will make a
few observations. First, as mentioned earlier, because we are ultimately interested in the
average angular structure function, we can ignore factors in the resummed cross section that
are independent of Ga or the resolution parameter R. Thus, we will not discuss nor resum the
hard function nor the unmeasured jet and soft functions. Also, we will only consider a single
measured jet in an event. This prevents a study of inter-jet correlations of the angular
correlation function, but for this paper we are most interested in the intra-jet dynamics.
Anyway, the existence of factorization of jet observables essentially trivializes correlations
between jets since it implies that correlations can only come from the soft function. From
these observations, we only need to resum the measured jet and soft functions of a single
jet.
With these considerations, we will need to compute the convolution between the measured
jet and soft functions:
dσ
dGa∝∫dG ′a J(Ga − G ′a;µJ , µ)S(G ′a;µS, µ) , (42)
where µJ and µS are the jet and soft scales respectively. We refer the reader to [18] for
the details of generic NLL-level resummation. Here, we will use the results collected there
appropriate for the angular correlation function. The resummed differential cross section for
the angular correlation function of a single measured jet at NLL is
dσ
dGa∝(µJω
)aωJ
(µS tana−1 R
2
ω
)ωS
[1 + fJ(Ga) + fS(Ga)]eKJ+KS+γE(ωJ+ωS)
Γ(−ωJ − ωS)
[1
G1+ωS+ωJa
]
+
.
(43)
ω is the − component of the jet’s momentum and γE is the Euler-Mascheroni constant. The
functions ωJ , ωS, KJ , KS, fJ and fS are written in detail in Appendix B. They depend on
the jet and soft scales and the factorization scale µ. The jet and soft scales will be in general
sensitive to the value of Ga and the resolution parameter R. At very small values of Ga, the
resummed distribution can become negative and in general will need to be matched onto
a non-perturbative shape function in that region. We do not attempt to correct the shape
at very small Ga and instead just set the cross section to zero where it would otherwise be
negative.
16
The average angular correlation function can then be computed from the cross section in
Eq. 43 by integrating over Ga:
〈Ga(R)〉 =
∫ Gmaxa
0
dGadσ
dGaGa
∝∫ Gmax
a
0
dGa(µJω
)aωJ
(µS tana−1 R
2
ω
)ωS
eKJ+KS+γE(ωJ+ωS)
Γ(−ωJ − ωS)
[1 + fJ + fS]
GωS+ωJa
, (44)
where Gmaxa =
tana R2
4is the maximum value of the angular correlation function for a jet with
two constituents. We choose the scales µJ and µS so as to eliminate the logarithms that
remain in the resummed distribution. The choice of these scales can be seen from the form
of the fJ and fS terms as given in the appendix. We find
µJ = ωG1/aa , µS =ωGa
tana−1 R2
. (45)
With this choice of scales, the average angular correlation function simplifies:
〈Ga(R)〉 ∝∫ Gmax
a
0
dGaeKJ+KS+γE(ωJ+ωS)
Γ(−ωJ − ωS)[1 + fJ + fS] . (46)
Note, however, that there is non-trivial dependence on Ga in the functions ωJ , ωS, KJ and
KS. Finally, to determine 〈∆Ga〉, we compute
〈∆Ga〉 =d log〈Ga〉d logR
. (47)
Plots of the average angular structure function as computed in SCET and compared to
Monte Carlo and NLO corrections will be presented in the following sections.
1. Lowest-Order Expansion
Before continuing, it is illuminating to expand the angular correlation function to lowest
order in the coupling αs. To do this, we will need to expand Eq. 43 to O(αs). The form of all
of the functions in Eq. 43 are given in Appendix B and, in particular, the expansions of the
Gamma, harmonic number and polygamma functions are needed. The necessary expansions
are given in the appendix. To leading order in αs, we find
dσ
dGa∝ αs(µ)
2πT2i
[4 log tan
R
2− 4
alog Ga −
1
a
(ci + log
tan2 R2
tan2 R0
2
)]1
Ga+O(α2
s) , (48)
17
where the factor ci depends on the flavor of the jet:
cq =3
2, cg =
β02CA
. (49)
To compute this, we have set the jet and soft scales so as to minimize the logarithms that
appear in the cross section as defined in Eq. 45. In this expression, note that the non-cusp
piece of the anomalous dimension of the measured jet functions appears in the term in
parentheses.
From this expression for the cross section differential in the angular correlation function,
we integrate over Ga to compute the average angular correlation function. To O(αs) we find
〈Ga〉 ≡∫ tana R
24
0
dGadσ
dGaGa
∝∫ tana R
24
0
dGaαs(µ)
2πT2i
[4 log tan
R
2− 4
alog Ga −
1
a
(ci + log
tan2 R2
tan2 R0
2
)]1
GaGa
=αs(µ)
2πT2i
tana R2
a
(1 + log 4− ci
4− 1
4log
tan2 R2
tan2 R0
2
)+O(α2
s) . (50)
Any overall factor independent of R does not affect the average angular structure function
because
〈∆Ga(R)〉 ≡ d log〈Ga〉d logR
=R
〈Ga〉d〈Ga〉dR
. (51)
To lowest order, the average angular structure function is independent of αs and only de-
pendent on the color of the jet through the ci term. Eq. 50 results in the average angular
structure function of
〈∆Ga(R)〉 =R
sinR
a−
2
4 + 4 log 4−(ci + log
tan2 R2
tan2R02
)
+O (αs(µ)) . (52)
Eq. 52 contains much of the physics that we expect affects the form of the angular
structure function. The naıve expectation for 〈∆Ga〉 is 〈∆Ga〉 ' a. Eq. 52 contains an
O(1) correction to this result that is negative. This was interpreted in [12] as an effect
due to the running coupling. However, here, this is probably not the source of this effect
because even for fixed coupling the negative term exists. This is instead probably due to
SCET itself because only collinear and soft emissions are included with respect to full QCD.
Including all terms in the resummed result should decrease the average angular structure
18
function further due to both the running coupling and because an arbitrary number of soft
and collinear emissions are considered.
Also, note that the term ci is larger for quarks than for gluons with sufficiently many
flavors of quarks:
cq =3
2≥ 11
6− Nf
9= cg , (53)
for Nf ≥ 3. This implies that, for sufficiently many flavors, 〈∆Ga〉g > 〈∆Ga〉q, an observation
that was also made in [12]. There, this was attributed to the fact that gluons have more
color than quarks and so radiate more at larger angles, effectively decreasing the strength of
the collinear singularity with respect to quarks. We expect that the resummation magnifies
the distinction between quarks and gluons.
Another interesting observation to be made about the form of the angular structure
function is that, to this order, it is Lorentz invariant. We then expect that all jets, regardless
of energy (so long as it is above the hadronization scale of QCD), have an angular structure
function that deviates only slightly from the form in Eq. 52. In particular, note that Eq. 52
is the infinite jet energy limit of the (all-orders) angular structure function. The contribution
of higher orders to the angular structure function would contain prefactors of αs(µ) which
would vanish as µ → ∞. If we ignore the finite R terms from the expansion of sine and
tangent, 〈∆Ga(R)〉 is very flat, signifying very near scale invariance over a large dynamical
range R. Flatness is only broken by a term that goes like 1/ log RR0
which is only important
at very small R/R0.
It is accurate, then, to represent the angular structure function in the form (again, ig-
noring the finite R terms from sine and tangent)
〈∆Ga(R)〉 ' a− γASF , (54)
where γASF might be called the anomalous dimension of a QCD jet and is independent of
a. This anomalous dimension is a robust quantity that is intrinsic to the flavor of the jet
and properties of QCD. Measuring this property of the angular structure function in data
would be very interesting. It is important to note, however, for all of the above comments,
O(αs) contributions to the average angular structure function have been ignored. These are
expected to be comparable in size to the second term in Eq. 52. Note in particular that
NNLO contributions can be just as, or even more, important than the contributions from
19
resummation. Indeed, for jets with three constituents, it was computed in [12] that the
effect at this order is to increase the average angular structure function.
F. Non-Perturbative Physics Effects
In addition to the perturbative physics contribution to the angular structure function, we
would also like to understand the effects from non-perturbative physics. For jets produced
in an e+e− collider, the dominant non-perturbative effect is from hadronization. A simple
physical argument can be used to determine how hadronization affects the angular structure
function. The partons created from the parton shower will be connected to one another
by color strings which stretch across the event. After the termination of the parton shower
at an energy scale of about 1 GeV, these color strings are allowed to break to create a
quark-antiquark pair if it is energetically favorable. This string breaking continues until all
particles are connected by strings with sufficiently low tension and are then associated into
hadrons. In the process of breaking the strings and creating quark pairs, the number of
particles that are created at small angles with respect to one another increases from that
which was created in the perturbative parton shower. Thus, hadronization increases particle
production at small angles, effectively increasing the strength of the collinear singularity and
decreasing the value of the average angular structure function.
The effect of hadronization decreasing the average angular structure function can also be
quantitatively studied. Note that the angular correlation function is just the (squared) mass
of a jet from constituents that are separated by angular scale R or less. Dasgupta, Magnea
and Salam [30] studied the effect of non-perturbative physics on the transverse momentum
and mass distributions of jets at hadron colliders. For the mass, they found that the leading
correction due to hadronization is
〈δM2〉 ∼ CR0 +O(R30) , (55)
where C is independent of R0, the jet radius. For the angular correlation function, we
expect that the effect of hadronization would also result in a correction proportional to R,
the resolution parameter of the angular correlation function. We can write
〈Ga(R)〉 ' CpertRa + Cnon-pertR , (56)
20
where Cpert is the perturbative contribution to the angular correlation function and Cnon-pert
is the non-perturbative contribution. The average angular structure function that follows
from this is
〈∆Ga(R)〉 =aCpertR
a + Cnon-pertR
CpertRa + Cnon-pertR< a , (57)
where the inequality follows when a > 1. Note that the perturbative angular structure
function is approximately a and so, indeed, hadronization effects decrease the value of the
angular structure function.
The argument presented here and in [30] relies on the one-gluon approximation to de-
termine the effect of hadronization. Universality of the hadronization and power correction
effects was argued with the one-gluon approximation in refs. [31, 32] and demonstrated for
event shapes in SCET in refs. [33, 34]. The arguments in refs. [33, 34] relied on the boost
invariance of the soft function for back-to-back jets. How the argument might extend to an
arbitrary number of jets in arbitrary directions is unclear as the boost invariance is, at least
naıvely, broken. We will not discuss how this might be extended, but we note that, because
of the qualitative and quantitive arguments from the one-gluon approximation, we expect
that the universality holds in SCET.
G. The Angular Correlation Function at the LHC
Finally, we will discuss how the results obtained here for the SCET resummation might
be extended to the LHC, to processes initiated by pp collisions. For an observable O that
factorizes at hadron colliders, the cross section can be written in the schematic form [35–37]
dσ
dO = H(µ)× CabBa(µ)Bb(µ)⊗[∏
ni
Jni(O;µ)
]⊗ S(O;µ) (58)
The beam functions Bi encode the properties of the initial parton i and the matrix Cab
weights the colliding partons by the appropriate cross section. Indices a and b are implicitly
summed over. In this case, the flavor of the jet functions depends on the flavor of the initial
colliding partons which affects the admixture of quark and gluon jets that contribute to
O. Note also that the soft function includes contributions from radiation from initial state
partons. Therefore, while not necessarily manifest in Eq. 58, the beam functions implicitly
affect the jet and soft functions.
21
Nevertheless, we expect that the angular correlation function has nice factorization prop-
erties at hadron colliders. With the goal of computing the average angular structure function,
we can again ignore anything in the factorization of the angular correlation function that is
independent of Ga or the resolution parameter R:
dσ
dGai∝ CabBa(µ)Bb(µ)⊗ Jni
(Gai;µ)⊗ Sna,nb;n1···nN(Gai;µ) , (59)
where we have chosen to measure Ga in jet i in an event with N jets. Dependence on the
beam functions has been retained, however. This is because, for a given set of jets 1, . . . , N ,
different initial states contribute to the cross section with different weights. Thus, the beam
function contribution to the factorization, CabBa(µ)Bb(µ), is actually not an overall constant
factor and so must be included. Note also that the color of the colliding partons affects the
radiation included in the soft function. The beam functions are universal and so can be
computed once and for all. While this is not a rigorous proof of factorization of the angular
correlation function, many of the results obtained in the e+e− collider context should be
able to be recycled for the hadron collider case. This deserves significant future study.
III. COMPARISON TO FIXED-ORDER CALCULATION
In this and the following section, we will focus most of our attention on the (proper)
angular correlation function with a = 2:
G2(R) ≡ G(R) =1
2E2J
∑
i 6=jEiEj sin θij tan
θij2
Θ(R− θij) . (60)
To evaluate this jet observable in SCET, we must choose the hard, jet and soft scales. For
many of the comparison plots we choose the following scales:
µH = ω , µJ = ωG1/2 , µS =ωG
tan R2
. (61)
These choices of scales minimize logarithms that appear in the resummed distribution. How-
ever, it is important to understand the dependence of the result on the choice of these scales
and so we will also present plots in which the scales are varied by the standard factors of
2 and 1/2. The evaluation of the average angular structure function from the SCET cross
section is done numerically. Note that for consistency of the factorization the jet scale µJ
must be larger than the soft scale µS which is the requirement that
tanR
2� G1/2 . (62)
22
0.2 0.4 0.6 0.8 1.0 1.2 1.4R
0.5
1.0
1.5
2.0
2.5XDGHRL\
(a) SCET vs Pythia8
0.2 0.4 0.6 0.8 1.0 1.2 1.4R
0.5
1.0
1.5
2.0
2.5
3.0XDGHRL\
(b) Hard scale variation
0.2 0.4 0.6 0.8 1.0 1.2 1.4R
0.5
1.0
1.5
2.0
2.5
3.0XDGHRL\
(c) Jet scale variation
0.2 0.4 0.6 0.8 1.0 1.2 1.4R
0.5
1.0
1.5
2.0
2.5
3.0XDGHRL\
(d) Soft scale variation
FIG. 3. Plots of the average angular structure function for quark (red) and gluon (blue) jets.
Fig. 3(a) compares the curves from SCET resummation (solid) to anti-kT jets from Pythia8
(dashed). The Pythia8 curves were computed from 3 jet final states in which all jets had equal
energy. Figs. 3(b), 3(c) and 3(d) compare the Pythia8 curves to SCET bands in which the hard,
jet and soft scales have been varied by a factor of 2. To make these curves, the jet radius has been
set to be R0 = 1.0 and the energy of the jets is 300 GeV.
To maintain this separation, the resolution parameter cannot be too small; we will only
consider R & 0.1. For smaller values of R, logarithms of R become large and must be
resummed, which is beyond the scope of this paper.
The average angular structure function as computed in SCET is plotted in Fig. 3 where
the curves for quark and gluon jets are compared to the output of Pythia8. The Pythia8
curves will be discussed in the next section. Fig. 3(a) compares the quark and gluon curves
with the hard, jet and soft scales set to their values in Eq. 61. The observations from
the previous section are apparent with the quark average angular structure function less
23
than the gluon average angular structure function and both slightly less than 2. The scale
variations of these curves are shown in Figs. 3(b), 3(c) and 3(d). Note that in particular
there is relatively wide range over which the angular structure function varies when the jet
and soft scales are changed by a factor of 2.
Because the average angular correlation function is defined by integrating over the entire
range of Ga, its value and shape is sensitive to radiation in all regions of phase space.
Resummation is necessary for an accurate description of the physics in the singular regions
of phase space while higher fixed-order contributions are necessary for a good description
in the non-singular regions of phase space. A proper treatment of resummation and fixed-
order involves consistently matching the two contributions so that the resulting distribution
is accurate order by order in αs over the entire phase space. This matching is a non-trivial
procedure and, instead, we will just focus on the contribution from higher fixed-order matrix
elements. This will give us a sense, at least, for how fixed order and resummation affect the
average angular structure function.
To do this, we use NLOJet++ v. 4.1.3 [38, 39], based on the dipole subtraction method
of [40], to compute the average angular structure function to NLO in e+e− collisions. NLO-
Jet++ can compute matrix elements to NLO for up to 4 final state partons (and, at tree
level, up to 5 final state partons) and so, by demanding jet requirements, produces jets with
very few constituents in them. This results in very inefficient calculation of cross sections.
Also, the public version of NLOJet++ does not record flavor information of partons so the
identity of quark and gluon jets cannot be easily determined. Further, it is not enough
that the cross section differential in the angular correlation function at fixed R is smooth
for the average angular structure function to be smooth. The distributions must also be
smooth over R so that the derivative that defines the average angular structure function is
well-behaved. To assuage these issues, in this section, we will define an event-wide angular
correlation function, where the sum in Eq. 5 runs over all particles in the event.
The event-wide angular correlation function is defined over all particles in the event with
no jet algorithm cut. In the limit that there are three final state particles this reduces
precisely to the angular correlation function of the hardest jet, extending up to an R of
about the radius of the hardest jet. The angular correlation function will only include the
contribution from the two closest partons because the third parton must be very far away
in angle. This argument doesn’t hold at higher orders, but for those cases we expect that
24
0.2 0.4 0.6 0.8 1.0 1.2 1.4R
0.5
1.0
1.5
2.0
2.5
3.0XDGHRL\
FIG. 4. Comparison of NLOJet++ calculation of the event-wide average angular structure function
in 3 jet final states to NLO (dotted) to SCET NLL resummation of the average angular structure
function for quark jets (solid). Two curves from Pythia8 are shown: the dashed curve is the
average angular structure function for quark jets from e+e− → 3 jets and the dot-dashed curve is
the average angular structure function from e+e− → 2 jets.
the event-wide definition will be an average over the angular correlation functions of quark
and gluon jets.
We present the calculation of the average angular structure function from NLOJet++
for three final state partons to NLO in Fig. 4. The center of mass energy is taken to be 600
GeV in e+e− collisions. At a e+e− collider, most of the time, the hardest jet will contain a
quark and a radiated gluon so we compare the output of NLOJet++ to NLL resummation
results for quark jets. Fig. 4 also contains two curves of quark jets from Pythia8 which will
be discussed in the next section. The NLO calculation of the angular structure function is
approximately flat and greater than 2 which we interpret as an effective weakening of the
collinear singularity due to the presence of wide-angle radiation. The fact that the NLO
result is slightly larger that 2 was anticipated in [12] where it was shown that a jet with
three constituents should have an average angular structure function larger than 2 by a term
proportinal to αs. Matching the calculations from NLL and NLO would produce a curve
that interpolates between the NLL result at small R and the NLO result at large R.
To generate the NLOJet++ curve, about one trillion events were processed over about
1 CPU year. Even with this many events, the average angular structure function from
NLOJet++ is still quite noisy. However, the noise can be reduced by averaging the angular
structure function over a small range in R at each point. This was done for the curve in
25
Fig. 4. Computing the average angular structure function in 4 jet final states to NLO was
attempted in the same CPU time as the 3 jet results. However, the resulting curves were
much too noisy to be used. To produce curves at higher orders using NLOJet++ probably
requires centuries of CPU time for distributions to converge. However, other programs such
as BlackHat [41] might be better-suited to higher multiplicity final states at NLO. Work in
this direction is ongoing.
IV. PARTON SHOWER MONTE CARLO COMPARISON
In this section, we compare our calculation of the average angular structure function
from SCET to the output of Monte Carlo event generator and parton shower. Through the
Sudakov factor which dictates the probability that no branchings occur between two scales
of an evolution variable, the parton shower resums logarithms of the evolution variable
that arise from soft and collinear emissions. Monte Carlo generators create fully exclusive
events and so the process of resummation of the logarithms is distinct from that in SCET,
for example, and examining the differences is interesting. The parameter that defines the
evolution in the parton shower is also (relatively) arbitrary and different choices of the
evolution variable lead to different emphases on soft or collinear splittings. In addition,
hadronization and other non-perturbative physics is described by phenomenological models
which can be used to understand the size and effect of power-suppressed contributions to
observables. All of these points and their effects will explored in this section.
For most of the Monte Carlo comparison, we first generated tree-level events for the
process e+e− → qqg using MadGraph5 v. 1.4.5 [42] at center-of-mass energy of 900 GeV.
These partons are required to each have equal energy E = 300 GeV so that they are well-
separated and factorization-breaking terms in the soft function are minimized. These events
were then showered using the pT -ordered shower of Pythia8 v. 8.162 [43]. All default settings
of Pythia8 were used except for turning hadronization on and off to study the difference.
In most plots hadronization in Pythia8 has been turned off. To study the effect of using
different evolution variables in the parton shower we shower the MadGraph events with
VINCIA v. 1.0.28 [44]. From the showered events, jets were found with the FastJet v. 3.0.2
[45] implementation of the anti-kT algorithm [46]. We choose the jet radius to be R0 = 1.0.
The three hardest jets are required to have energy between 250 and 350 GeV and we identify
26
jets as coming from a quark or gluon by demanding that the cosine of the angle between
the jet axis and the direction of a parton from MadGraph be greater than 0.9.
In Fig. 3, we plot the average angular structure function for quark and gluon jets identified
in Pythia8 (with no hadronization) and the angular structure function as computed in SCET.
Note that the average angular structure function as computed in SCET is significantly
smaller than that from Pythia8, especially at larger R. This difference can be attributed to
higher order effects which were shown in the previous section to increase the value of the
average angular structure function. Fig. 3 also illustrates the distinction between quark and
gluon jets. For most of the range of 0 < R < 1, the average angular structure function for
gluon jets is greater than that for quark jets, reflecting the fact that gluons have more color
and radiate more at wider angles than do quarks. This effect is present in both the Pythia8
curves and the resummed calculation. Because the SCET calculation only included effects
from jets with at most two constituents, the curves terminate precisely at the jet radius of
R0 = 1.0. For these anti-kT jets in Pythia8, the edge effects from the jet algorithm are small,
extending only over a range of at most R = 0.8 to R = 1.2. Also, we have not plotted the
SCET curves below R = 0.1, where they begin to deviate substantially from their value at
larger R.
Fig. 4 compares the average angular structure function from NLOJet++ to quark jets
in SCET and two different curves from Pythia8. The different Pythia8 curves exhibit the
affect of wide angle radiation captured by the jet on the average angular structure function.
In that figure, the dashed curve is the quark jet average angular structure function from the
Pythia8 sample described above. The dot-dashed curve is the the average angular structure
function from quark jets from e+e− → qq samples generated and showered in (otherwise
default) Pythia8. The center of mass energy was set to be 600 GeV and the jets were
required to have energy within 50 GeV of 300 GeV. Higher order effects are obvious. The
Pythia8 curves agree well with one another at small R up to R ∼ 0.6 and then diverge at
larger R. The jets in the 3 jet sample collect wide-angle radiation from the neighboring jets
which increases the average angular structure function at large R. To fully understand the
rise within an analytic calculation requires matching fixed-order to resummed result. Fixed-
order contributions are responsible for the wide-angle emissions that increase the average
angular structure function because SCET factorization effectively decouples the jets.
As discussed in Sec. II F, we expect the effect from non-perturbative physics on the
27
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4R0.0
0.5
1.0
1.5
2.0
2.5
3.0XDGHRL\
(a) Identified gluon jets
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4R0.0
0.5
1.0
1.5
2.0
2.5
3.0XDGHRL\
(b) Identified quark jets
FIG. 5. Comparison of the average angular structure function as computed in Pythia8 with (dotted)
and without (dashed) hadronization.
angular structure function to be small and relatively well-understood. In particular, relying
on arguments from the one-gluon approximation, we expect that hadronization increases the
strength of the collinear singularity and that this effect is most prominent at small values of
R. In Fig. 5, we have plotted the average angular structure function for quarks and gluons
comparing the curves with hadronization turned on or off. Indeed, the effect is small but
unambiguous: hadronization effectively increases the strength of the collinear singularity.
As discussed earlier, extending the arguments from [33, 34] on the effect of non-perturbative
physics would be greatly desired to fully describe (at least) average behavior of hadronization
for events with multiple jets.
A. Monte Carlo Error Estimates
Finally in this section, we would like to get a handle on the error or uncertainty in
the Monte Carlo parton shower in Pythia. Typically, this is done by studying the output
of different tunes of the same Monte Carlo program or comparing different Monte Carlo
programs altogether. In particular, as is relevant for the parton shower, the evolution
variable of the parton shower dictates when and how emissions should occur. Ref. [47]
observed differences in event shape variables as computed in Pythia 6.4 [48] between two
tunes; one pT -ordered and the other virtuality ordered. However, these two tunes had
other distinctions as well and so purely the effect of the evolution variable is obscured. Also,
28
0.2 0.4 0.6 0.8 1.0 1.2 1.4R
0.5
1.0
1.5
2.0
2.5
3.0XDGHRL\
(a) Identified gluon jets
0.2 0.4 0.6 0.8 1.0 1.2 1.4R
0.5
1.0
1.5
2.0
2.5
3.0XDGHRL\
(b) Identified quark jets
FIG. 6. Comparison of SCET computation (solid) and Pythia8 (dashed) of average angular struc-
ture function to the output of VINCIA Monte Carlo parton shower with two different evolution
variables: pT and virtuality. The shaded region lies between the curves from VINCIA.
comparing two different Monte Carlos is subtle because the number of differences is typically
huge and so isolating effects of single parameters or choices is very difficult.
Here, we would like to study the effect of different evolution variables in the parton
shower. The choice of the evolution variable is only a change of variables in the Sudakov
form factor and so must produce the exact same leading-log resummation for any (consistent)
choice of evolution variable. However, the choice of evolution variable can lead to higher
log-order effects through the scale at which αs is evaluated or by emphasizing soft over
collinear splittings, for example. To study the differences, we use the VINCIA [44] parton
shower plug-in for Pythia8 which is based on 2-to-3 splittings as opposed to the standard
1-to-2 splittings as in Pythia and Herwig [49]. VINCIA includes a flag which allows the user
to change only the evolution variable. For concreteness, we will consider pT -ordering and
virtuality ordering.
In Fig. 6 we have plotted the SCET resummation and Pythia8 output for the average
angular structure function as well as a band which extends over the range between the
output of the pT -ordered and the virtuality ordered shower in VINCIA. The exact same
requirements on the jets were made in the VINCIA sample as in the Pythia8 sample as
described earlier. Over most of the range in R, the lower edge of the VINCIA band is set
by the pT -ordered shower and upper edge by the virtuality ordered shower. This is expected
as the pT -ordered shower emphasizes collinear emissions more than the virtuality-ordered
29
shower. Note also that the band is slightly above the output of the pT ordered shower in
Pythia8. Part of this effect could be due to the default matrix element matching in VINCIA:
final states with up to 5 partons are matched to tree-level matrix elements. Regardless of
the details, the effect of changing the evolution variable is large. Understanding if and how
parton showers resum higher order logarithms with different evolution variables, matching
schemes, etc., is necessary to understand the source of the differences.
V. CONCLUSIONS
The average angular correlation and structure functions capture the average scaling prop-
erties of QCD jets. We have presented a calculation of the angular correlation function to
NLL accuracy in SCET and compared this result to the Pythia8 Monte Carlo parton shower
and to fixed-order results from NLOJet++. Comparing the resummed SCET result to the
fixed-order NLOJet++ result provides a good understanding as to the behavior of the parton
shower result. However, for a full understanding, matching of the resummed and fixed-order
distributions is required. Much like the jet shape [15], the average angular structure function
could be used for tuning of the Monte Carlo. Because it is a two-point correlation function,
the angular correlation function captures distinct information from the jet shape and so this
tuning would be non-trivial.
There are several directions for extending the study presented here. First, it would be
desirable to compute the angular correlation function in collisions at the LHC. It remains an
outstanding problem to use SCET to resum logarithms for arbitrary observables in hadron
colliders because factorization of the (colored) initial and final states is highly non-trivial.
However, using the observations from Sec. II G, the computation of the average angular
structure function at hadron colliders might only require a reinterpretation of the results
presented here. Recently, NLO results were obtained for pp → 4j events [50] from which
any IRC-safe observable could be computed. In particular, for four final state partons,
the hardest jet can contain up to three constituents which would be beyond the resummed
order in the SCET calculation. Also, at a hadron collider, underlying event or pile-up
produce significant background radiation that can be collected into a jet. A procedure
to determine the contribution to a jet from these non-perturbative sources is necessary to
properly determine jet energy scales and to study substructure. The results presented here
30
could be used to determine the average contribution to a jet using the procedure introduced
in [12].
For a more accurate prediction of the angular correlation function, matching of NLL and
NLO results must be done to have good control of the distribution over the entire phase
space. Factorization of jet observables allows for a process-independent computation of
the NLL resummed result; however, the fixed-order calculation is process dependent and
must be couched in a particular study. We showed that the average angular structure
function is sensitive to wide-angle radiation so matching is vital for accurate predictions.
NLOJet++ or results like those from [50] are promising in their applicability to generic
processes. It is unlikely that QCD jet observables can be reliably computed to NNLL or
beyond analytically because non-global logarithms become important. Nevertheless, by
studying limiting behavior such as in [6] the effect of these non-global logarithms might be
reduced.
Finally, as there exist few jet substructure observables that have been (or even can be)
computed with analytic methods, it is important to compute those that are possible. The
calculation of the angular correlation function provides powerful insight into the behavior
of QCD and the dynamic properties of jets. Though scale-invariance is broken in QCD by a
running coupling, jets maintain a fractal, conformal structure to very good approximation
over a wide dynamical range.
Appendix A: Measured Jet Functions
Here, we present the finite pieces of the measured jet functions for quark and gluon jets
as defined by a kT -type algorithm for the angular correlation function. These functions are
composed of contributions from δ-functions and +-distributions. For a function g(x), we
define the +-distribution as [51]
[g(x)Θ(x)]+ = g(x)Θ(x)− δ(x)
∫ 1
0
dx′ g(x′) , (A1)
so that ∫ 1
0
dx [g(x)Θ(x)]+ = 0 . (A2)
From this definition, it is straightforward to compute the measured jet functions. The terms
that are infinite in four dimensions were presented in Sec. II B. The terms that are finite in
31
four dimensions are, for a quark jet:
Jqω(Ga, ε0) =αsCF
2π
{(13
2− 9a− 8
12(a− 1)π2 +
3
2log
µ2
ω2 tan2 R0
2
+a/2
a− 1log2 µ
2
ω2
+ logµ2
ω2log
tan2 R2
tan2 R0
2
+1
2log2 tan2 R0
2+a− 1
2log2 tan2 R
2
)δ(Ga)
+
Θ(Ga)Θ(Gmax
a − Ga)
4
a− 1
log GaGa
− 2
a− 1
log µ2
ω2 tan2(a−1) R2
Ga
+4
a
1
Galog
1 +√
1− 4Gatana R
2
1−√
1− 4Gatana R
2
− 3
a
√1− 4Ga
tana R2
Ga
+
. (A3)
For a gluon jet, the finite terms are:
Jgω(Ga, ε0) =αs2π
{(67
9CA −
23
9nFTR −
9a− 8
12(a− 1)π2 +
β02
logµ2
ω2 tan2 R0
2
+CAa/2
a− 1log2 µ
2
ω2+ CA log
µ2
ω2log
tan2 R2
tan2 R0
2
+1
2log2 tan2 R0
2
+a− 1
2log2 tan2 R
2
)δ(Ga) +
Θ(Ga)Θ(Gmax
a − Ga)
−β0
a
√1− 4Ga
tana R2
Ga
+4CAa
1
Galog
1 +√
1− 4Gatana R
2
1−√
1− 4Gatana R
2
+2CA − 4nFTR
3a tana R2
√1− 4Ga
tana R2
+4CAa− 1
log GaGa
− 2CAa− 1
log µ2
ω2 tan2(a−1)R2
Ga
+
. (A4)
Gmaxa is the largest value that Ga can take for a jet with two constituents:
Gmaxa =
tana R2
4, (A5)
where we have taken the leading λ dependence for the collinear modes.
Appendix B: Resummed Distribution for Angular Correlation Function
The expression for the resummed cross section is
dσ
dGa∝(µJω
)aωJ
(µS tana−1 R
2
ω
)ωS
[1 + fJ(Ga) + fS(Ga)]eKJ+KS+γE(ωJ+ωS)
Γ(−ωJ − ωS)
[1
G1+ωS+ωJa
]
+
.
(B1)
32
ωJ , ωS, KJ and KS result from the resummation of the individual jet and soft functions
[52–56]. The functions ωJ and ωK are defined by ωJ ≡ ωF (µ, µJ) and ωS ≡ −ωF (µ, µS)
where
ωF (µ, µ0) = − 4T2i
(a− 1)β0
[log r +
(Γ1cusp
Γ0cusp
− β1β0
)α(µ0)
4π(r − 1)
]. (B2)
β0 is the coefficient of the one-loop β-function as defined in Eq. 30 and β1 is the two-loop
coeffcient:
β1 =34
3C2A −
10
3CANf − 2CFNf . (B3)
r is the ratio of the strong coupling at two scales:
r =αs(µ)
αs(µ0), (B4)
and the energy dependence of the strong coupling is given by the two-loop expression
1
αs(µ)=
1
αs(Q)+β02π
log
(µ
Q
)+
β14πβ0
log
[1 +
β02παs(Q) log
(µ
Q
)], (B5)
for αs evaluated at the two scales µ and Q. The terms Γ0cusp and Γ1
cusp are the one- and
two-loop coefficients of the cusp anomalous dimension. Their ratio is given by [57]
Γ1cusp
Γ0cusp
=
(67
9− π2
3
)CA −
10
9Nf . (B6)
The function KJ is given by
KJ(µ, µJ) = − γ0i2β0
log r − 8πaT2i
(a− 1)β20
[r − 1− r log r
αs(µ)
+
(Γ1cusp
Γ0cusp
− β1β0
)1− r + log r
4π+
β18πβ0
log2 r
],(B7)
with γ0i defined as
γ0i = 4T2i
(log
tan2 R2
tan2 R0
2
+ ci
), (B8)
where ci is defined in Eq. 49. KS is defined similarly
KS(µ, µS) =8πT2
i
(a− 1)β20
[r − 1− r log r
αs(µ)+
(Γ1cusp
Γ0cusp
− β1β0
)1− r + log r
4π+
β18πβ0
log2 r
].
(B9)
33
The functions fJ and fS are generated by the convolution of the jet and soft functions.
Accurate to NLL, they are
fJ(Ga;µ, µJ) =αs(µJ)T2
i
2πΘ (Gmax
a − Ga){
2a
a− 1log2 µJ
ωG1/aa
+ log2 tan2 R0
2
+ (a− 1) log2 tan2 R
2− 2ci log tan
R0
2+
1
a− 1
2
a
[π2
6− ψ(1)(−ωJ − ωS)
]
+
[ci + log
tan2 R2
tan2 R0
2
+2
a− 1H(−1− ωJ − ωS)
]
×[2 log
µJ
ωG1/aa
+1
aH(−1− ωJ − ωS)
]}, (B10)
and
fS(Ga;µ, µS) = −αs(µS)
π
T2i
a− 1
(
logµS tana−1 R
2
ωGa+H(−1− ωJ − ωS)
)2
+π2
6− ψ(1)(−ωJ − ωS)
]. (B11)
H(x) is the harmonic number function defined by
H(x) =
∫ 1
0
1− tx1− t dt , (B12)
and ψ(1)(x) is the trigamma function
ψ(1)(x) =
∫ ∞
0
te−xt
1− e−t dt . (B13)
Note that the logarithms in these functions can be minimized by choosing
µJ = ωG1/aa , µS =ωGa
tana−1 R2
. (B14)
For expansion of the resummed distribution, the following relations are needed:
H(−1− ε)2 − ψ(1)(−ε) = −π2
2+O(ε) , (B15)
H(−1− ε)Γ(−ε) = −1 + γEε+O(ε2) . (B16)
ACKNOWLEDGMENTS
A. L. thanks Jon Walsh for extensive correspondence on SCET and the computation of
jet and soft functions as well as detailed comments on a draft of this paper. A. L. also thanks
34
Michael Peskin for his insisting that this work be done and helpful comments, especially in
comparing calculation to Monte Carlo. This work is supported by the US Department of
Energy under contract DE–AC02–76SF00515 and partial support by the U.S. National Sci-
ence Foundation, grant NSF–PHY–0969510, the LHC Theory Initiative, Jonathan Bagger,
PI.
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